We develop a combinatorial theory of vector bundles with connection that is natural with respect to appropriate mappings of the base space. The base space is a simplicial complex, the main objects defined are discrete vector bundle valued cochains and the main operators we develop are a discrete exterior covariant derivative and a combinatorial wedge product. Key properties of these operators are demonstrated and it is shown that they are natural with respect to the mappings referred to above. We also formulate a well-behaved definition of metric compatible discrete connections. A characterization is given for when a discrete vector bundle with connection is trivializable or has a trivial lower rank subbundle. This machinery is used to define discrete curvature as linear maps and we show that our formulation satisfies a discrete Bianchi identity. Recently an alternative framework for discrete vector bundles with connection has been given by Christiansen and Hu. We show that our framework reproduces and extends theirs when we apply our constructions on a subdivision of the base simplicial complex.

翻译：我们发展了一种在基空间适当映射下具有自然性的带联络离散向量丛的组合理论。基空间为单纯复形，主要定义的对象是离散向量丛值上链，主要建立的算子是离散外协变导数与组合楔积。这些算子的关键性质得到证明，并表明它们相对于上述映射具有自然性。我们还给出了度量兼容离散联络的良好定义，并刻画了何时离散向量丛具有平凡化或存在平凡低秩子丛的条件。利用这一机制将离散曲率定义为线性映射，并证明我们的形式满足离散比安基恒等式。近期，Christiansen与Hu提出了带联络离散向量丛的另一种框架。我们证明，当将我们的构造应用于基单纯复形的细分时，我们的框架复现并扩展了他们的结果。